S-wave conversion in marine surveys

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 13.1a

In a marine survey, the water depth is 100 m and a reflector is 3 km below the seafloor. Use Figure 13.1a to determine the optimum range of offsets for S-wave generation. Take Poisson’s ratio just below the seafloor as 0.35 and the P-wave velocity as 2.8 km/s. The velocity in sea water is 1.5 km/s.

Solution

Referring to Figure 13.1a(i), we see that the offset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x for the 3-km reflector is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x=2(0.1\times \tan \theta _{1} +3\times \tan \delta_{2} )=0.2\tan \theta _{1} +6\tan \delta_{2} , \end{align} (13.1a)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} is the angle of incidence of the P-wave at the bottom of the water layer, and $ \delta _{2} $ is the angle of refraction of the converted S-wave.

To find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{2} for a given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} , we need the S-wave velocity $ \beta _{2} $. Using equation (1,8) in Table 2.2a, we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta /\alpha =0.48 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma =0.35 , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{2} =0.48\alpha _{2} =0.48\times 2.8=1.3\ {\rm km/s} . To get the angles of incidence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} and of refraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{2} , we have from equation (3.1a),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \theta _{1} /1.5=\sin \delta_{2} /1.3\; ,\; \delta_{2} ={\rm sin}^{-1} \left(0.87\sin \theta _{1} \right) . \end{align}

Figure 13.1a.  PSSP reflections generated by conversion at the sea floor (after Tatham and Stoffa, 1976). (i) Geometry; (ii) the conversion coefficient versus angle of incidence in the water; the curves are labeled with the seafloor P-wave velocity in km/s. [Note: Conversion coefficient = (amplitude of converted wave/amplitude of incident wave.)

The optimum angles of incidence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} for S-wave conversion for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha =2.8\ {\rm km/s} are obtained by interpolating between the curves for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha =2.5 and 3.0 in Figure 13.1a(ii). This gives a range of about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 38^{\circ} to $ 80^{\circ } $ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} . The corresponding values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{2} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 32^{\circ} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 59^{\circ} . Equation (13.1a) now gives offsets of 4 and 11 km for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{1} =38^{\circ} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 80^{\circ} .

Problem 13.1b

Most marine S-wave surveys that wish to deal with S-waves utilize conversion at the reflector and recording with three-component geophones laid on the seafloor (OBC, ocean-bottom cables), so that only one mode conversion is involved. Assume a ray leaving an airgun source at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10^{\circ} to the vertical in water 100 m deep, an increase in P-wave velocities from 1.5 km/s at the seafloor to 3.0 km/s at a reflector 3 km below the sea floor (average velocity in the sediments of 2.25 km/s). Take $ \sigma $ in the sediments as 0.3. What will be the source-geophone offset?

Solution

The P-wave gives an incident angle at the reflector of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta =\sin^{-1} [(3.0/1.5) \sin 10^{\circ} ]=20.3^{\circ} . Using equation 1,8 in Table 2.2a, the S-wave velocity here is about 0.53 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha or 1.59 km/s. Hence the angle of reflection is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta=\sin^{-1} (0.53\sin 20.3^{\circ} )=10.6^{\circ} . The average direction of the P-wave ray in the sediments is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin^{-1} [(2.25/1.5) \sin 10^{\circ} ]=15.1^{\circ} . With constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma in the sediments, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta /\alpha is constant and, hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin \delta/\sin \theta =0.53 . Hence, the average direction of the S-wave is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin^{-1} (0.53\sin 15.1^{\circ} =7.9^{\circ} ) . The offset is thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 100 \tan 10^{\circ} +3000\tan 15.1^{\circ} +3000\tan 7.9^{\circ} =1240\ {\rm m}. \end{align}

This is a very reasonable offset.

Continue reading

Previous section Next section
Acquisition direction for marine 3D surveys Equally inclined orthogonal geophones
Previous chapter Next chapter
3D methods Specialized applications

Table of Contents (book)

Also in this chapter

External links

find literature about
S-wave conversion in marine surveys
Retrieved from "http://wiki.seg.org/w/index.php?title=S-wave_conversion_in_marine_surveys&oldid=133645"